Let $R$ be a Noetherian, commutative ring and $M,N$ be finitely generated $R$-modules. 

**Question**: Do the sets of minimal primes of $\text{Ext}^i_R(M,N)$ (for a fixed pair of $M,N$) become periodic for $i\gg 0$? 

**Background/Motivation**: In this part I will assume $R$ local (I don't think it's significant for the question, but some statements will be awkward in general). I was motivated to ask after this [recent question](http://mathoverflow.net/questions/32137/the-associated-prime-ideals-of-exti-rm-n).

It is known that the asymptotic behavior of these $\text{Ext}$ is governed by the singularity of $R$. For instance, if $R$ is regular, then $\text{Ext}^i_R(M,N)=0$ for $i>\dim R$ because $R$ has finite global dimension equals to $\dim R$. 

If $R$ is a hypersurface, then it follows from Eisenbud's [original paper](http://www.jstor.org/pss/1999875) on matrix factorization that the minimal resolution of  $M$ becomes eventually periodic of period $2$ after $\dim R$ spots, so the answer in this case is also affirmative. 

With a little bit more work, one can show that the supports of $\text{Ext}^i_R(M,N)$ (in fact, the annihilators) become eventually periodic of period $2$ when $R$ is a local *complete intersection*. 

EDIT: As Karl's comment indicated, perhaps a more realistic question is:

**Question 2**: Is the union of all minimal primes of $\text{Ext}^i_R(M,N)$ for $i>0$ a finite set? 

One situation when the answer is trivially yes is when $\text{Spec}R$ has an isolated singularity. 

I think the answer is no (even for question 2) in general, but not sure how to construct an example.